Sophie, I think that the inequalities from this Lecture 27 are the "correct" ones. A nice way to remember which way the unit condition should go is to note that there the greater or equal side is of the form \$$f(-)\$$, both in \$$f(x) \otimes f(x') \leq f(x\otimes x')\$$ and in \$$I \leq f(I)\$$. These two inequalities are like the multiplication and the unit of a monoid: in the case of the multiplication, you have *two* things going in and *one* thing coming out, which corresponds to two \$$f\$$'s on the left and one on the right. The unit of a monoid is just a constant, which means that it has *no* thing going in and again *one* thing coming out; just like the \$$f\$$'s in \$$I\leq f(I)\$$.

In case that all this sounds a bit mysterious or confusing to you now, let me just say that it will become clear further along the course! (Assuming that lax monoidal functors will be covered.)